We report an extensive velocity resolved survey at 1' resolution of 158-mu-m [C II] fine-structure line emission in the central 50 pc of our Galaxy. Spectra were taken toward about 625 positions with a new imaging far-infrared spectrometer on board the NASA Kuiper Airborne Observatory. The [C II] emission, tracing the distribution of dense gas dissociated and partially ionized by far-ultraviolet photons, is strongest toward the rotating circumnuclear disk surrounding Sgr A West. A continuous bridge of [C II] line emission connects the Sgr A complex to the thermal radio filaments in the radio arc 10' north of the center, thus strongly suggesting a direct physical connection between the two. Outside of the nuclear region, there is a striking anticorrelation between [C II] intensity and the distribution of dense molecular material. The brightest emission preferentially occurs near the edges of the massive Galactic center molecular clouds. We conclude that these clouds are predominantly ionized by external ultraviolet photons, in part from the Sgr A West region itself and in part from OB stars that have relatively recently formed near the center. The new data suggest a physical connection between the center and the massive interstellar gas clouds in the surrounding 50 pc. The +50 km s-1 molecular cloud, for instance, is very likely located within 10 pc of the center. This cloud and the blueshifted cloud complex associated with the radio arc are probably feeding material into the center of the Galaxy.